Question

How do you multiply \[\left( {4{x^2} - 7x - 1} \right)\left( {6{x^2} + x - 7} \right)\]?

Answer 1

Hint: In the given expression we need to do multiplication of polynomials by binomials. Hence, to multiply the polynomials, first we need to multiply each term in one polynomial by each term in the other polynomial using distributive law. Then, we need to simplify the resulting polynomial by adding or subtracting the like terms.

Complete step by step solution:
Given,
\[\left( {4{x^2} - 7x - 1} \right)\left( {6{x^2} + x - 7} \right)\]
Multiplying polynomials is similar to multiplying regular numbers; each term from the first polynomial needs to get multiplied by each term in the second, and then these smaller products get added together to form the final product.
Multiplying polynomials is done the exact same as we do multiplication of three-digit numbers. We pair off each term in the left polynomial with each one in the right, creating smaller products; the sum of these smaller products will be our final product.
So, we have
\[\left( {4{x^2} - 7x - 1} \right)\left( {6{x^2} + x - 7} \right)\]
\[ \Rightarrow \left( {4{x^2} - 7x - 1} \right) \times \left( {6{x^2} + x - 7} \right)\]
Use distributive law and separate the first polynomial as:
\[ \Rightarrow \left[ {4{x^2} \times \left( {6{x^2} + x - 7} \right)} \right] + \left[ { - 7x \times \left( {6{x^2} + x - 7} \right)} \right] + \left[ { - 1 \times \left( {6{x^2} + x - 7} \right)} \right]\]
Multiply the monomials from the first polynomial with each term of the second polynomial as:
\[ = \left( {4{x^2} \cdot 6{x^2}} \right) + \left( {4{x^2} \cdot x} \right) + \left( {4{x^2} \cdot - 7} \right) + \left( { - 7x \cdot 6{x^2}} \right) + \left( { - 7x \cdot x} \right) + \left( { - 7x \cdot - 7} \right) + \left( { - 1 \cdot 6{x^2}} \right) + \left( { - 1 \cdot x} \right) + \left( { - 1 \cdot - 7} \right)\]
Multiplying the terms, we get:
\[ = 24{x^4} + 4{x^3} - 28{x^2} - 42{x^3} - 7{x^2} + 49x - 6{x^2} - x + 7\]
Now, we need to add the like terms, hence we get:
\[ = 24{x^4} - 38{x^3} - 41{x^2} + 48x + 7\]
Therefore, we get:
\[\left( {4{x^2} - 7x - 1} \right)\left( {6{x^2} + x - 7} \right) = 24{x^4} - 38{x^3} - 41{x^2} + 48x + 7\]

Note: It should be noted that the resulting degree after multiplying two polynomials will be always more than the degree of the individual polynomials. When multiplying polynomials, Distributive Law of multiplication is used twice when two polynomials are multiplied, then look for the like terms and combine them. This may reduce the expected number of terms in the product.